Question

Sketch an example that shows that the left-sum error bound $\left|E_{LEFT\left(n\right)}\right|\le |f(b)-f(a\left)\right|∆x$does not necessarily hold for functions $f$that fail to be monotonic on $[a,b].$

Short answer

Sketch a curve which explains the statement that," the left-sum error bound $\left|E_{LEFT\left(n\right)}\right|\le |f(b)-f(a\left)\right|∆x$does not necessarily hold for function $\text{'}f\text{'}$".

This sketch explains the required statement.

Step-by-step solution

Step 1. Given information

$\left|E_{LEFT\left(n\right)}\right|\le |f(b)-f(a\left)\right|∆x$.

Step 2. Consider a function 'f' which is not monotonic on interval a,b.

The objective is to sketch a curve which explains the statement that, "the left sum error bound $\left|E_{LEFT\left(n\right)}\right|\le |f(b)-f(a\left)\right|∆x$does not necessarily hold for function $\text{'}f\text{'}$".

For a monotonic curve, the bounds on left sum or right sum is given by the inequality $\left|E_{LEFT\left(n\right)}\right|\le \left|LEFT\left(n\right)-RIGHT\left(n\right)\right|=|f(b)-f(a\left)\right|∆x$.

Thus, for a curve which is not monotonic in interval $\left[a,b\right]$will not necessarily have a inequality to represent the bounds.

Step 3. Draw a sketch which is both increasing and decreasing in the interval a,b.

Thus, this sketch explains the required statement.